Techniques of Differential Topology in Relativity (CBMS-NSF Regional Conference Series in Applied Mathematics)

Section 1 sets the mathematical definitions and conventions used in the later sections. Spacetime is defined as a real, four-dimensional connected smooth Hausdorff manifold on which is defined a global smooth nondegenerate Lorentzian metric. In addition, it is assumed that spacetime is time-orientable, which is not too big a restriction since as the author remarks, one can always find a time-orientable twofold covering of spacetime. Jacobi fields are introduced also, with the goal of eventually using them to study maximal geodesics. Known to physicists as the equation of geodesic deviation, the author derives the Jacobi equation, the solutions of which form an 8-dimensional vector space of Jacobi fields.

In section 2, the author gives definitions that allow one to discuss causality and time ordering for curves on spacetime. Special types of non-smooth curves, called trips, which (piecewise) are future-oriented timelike geodesics, are used to define a chronological ordering of points on spacetime. Causal trips are more restrictive, in that the curves are causal geodesics. The chronological ordering is shown to imply causal ordering, and both orderings are shown to be transitive. This allows the partitioning of spacetime into chronologi